A Model for Computational Science Investigations

AiS ChallengeSTI 2004

Richard Allen

Computational Science? Computational science seeks to gain an

understanding of science through the use of mathematical models on supercomputers.

Computational Science involves teamwork

Computational Science

Complements, but does not replace, theory and experimentation in scientific research.

Experiment

Computation

Theory

Computational Science

Is often used in place of experiments when experiments are too large, too expensive, too dangerous, or too time consuming.Can be useful in “what if” studies; e.g. to investigate the use of pathogens (viruses, bacteria, fungi) to control an insect population. Is a modern tool for scientific investigation.

Computational Science

Has emerged as a powerful, indispensable tool for studying a variety of problems in scientific research, product and process development, and manufacturing. • Seismology • Climate modeling • Economics• Environment• Material research

• Drug design• Manufacturing • Medicine • Biology

Analyze - Predict

Example: Industry

First jetliner to be digitally designed, "pre-assembled" on computer, eliminating need for costly, full-scale mockup.Computational modeling improved the quality of work and reduced changes, errors, and rework.

www.boeing.com/commercial/777family/index.html

Example: Roadmaps of the Human Brain

Cortical regions activated as a subject remembers the letters x and r.Real-time MRI techno-logy may soon be incor-porated into dedicated hardware bundled with MRI scanners allowing the use of MRI in drug evaluation, psychiatry, & neurosurgical planning. www.itrd.gov/pubs/blue00/hecc.h

tml

Example: Climate Modeling

3-D shaded relief representation of a portion of PA using color to show max daily temperatures.Displaying multiple data sets at once helps users quickly explore and analyze their data. www.itrd.gov/pubs/blue00/

hecc.html

Computational Science Process

Real World Problem

Identify Real-World Problem: Perform background research, focus

focus on a workable problem. Conduct investigations (Labs), if

if appropriate. Learn the use of a computational tool: C++,

Java, StarLogo, Excel, Stella, and Mathematica.Understand current activity and predict future

behavior.

Working Model

Simplify Working Model: Identify and select factors to describe important aspects of Real World Problem; deter- mine those factors that can be neglected. State simplifying assumptions. Determine governing principles, physical laws. Identify model variables and inter-relationships.

Mathematical Model Represent Mathematical

Model: Express the Working Model in mathematical terms; write down mathematical equations or an algorithm whose solution describes the Working Model.

In general, the success of a mathematical model depends on how easy it is to use and how accurately it predicts.

Computational Model Translate Computational

Model: Change Mathema- tical Model into a form suit- able for computational solu- tion.

Computational models include languages, such as C++ or Java, or software, such as StarLogo, Stella, Excel, or Mathematica.

Results/Conclusions Simulate Results/Con-

clusions: Run “Computational Model” to obtain Results; draw Conclusions. Verify your computer program; use check

cases; explore ranges of validity. Graphs, charts, and other visualization tools are

useful in summarizing results and drawing conclusions.

Real World Problem Interpret Conclusions:

Compare with Real World Problem behavior.

If model results do not “agree” with physical reality or experimental data, reexamine the Working Model (relax assumptions) and repeat modeling steps.

Often, the modeling process proceeds through several “cycles” until model is“acceptable”.

Computational Science Process

Computational Science Investigations

A Computational science investigation should include An application - a scientific problem of interest

and the components of that problem that we wish to study and/or include.

Algorithm - the numerical/mathematical repre-sentation of that problem, including any numerical methods or recipes used to solve the algorithm.

Architecture – a computing platform and software tool(s) used to compute a solution set for the algorithm.

Example: A Falling Rock

Determine the motion of a rock dropped from a height H, above the ground with initial

velocity V.

Working Model

Governing principles: d = v*t and v = a*t.Simplifying assumptions:

Gravity is the only force acting on the body. Flat earth. No drag (air resistance). Model variables are H,V, g; t, s, and v.

Working Model (cont.)

Form a discrete-in-time model to determine the position and velocity of the rock above the ground at equally spaced times, t0, t1, t2, …, tn; e.g. t0 = 0 sec, t1 = 1 sec; t2 = 2 sec, etc.

v0 v1 v2 … vn

s0 s1 s2 … sn

|______|______|____________|_____ t0 t1 t2 … tn

100

90

80

70

60

50

40

30

20s = D

ispl

acem

ent (

in m

eter

s)

2

0

0

0

Initial conditions

9.8 / sec , 1.0 sec100 , 10.0 / sec

Problem variables:0

10010.0

g m tH m V m

ts Hv V

10

0

0 1 2 3 4t = time (in seconds)

An Illustration

An Illustration (cont.)

100

90

80

70

60

50

40

30

20s = D

ispl

acem

ent (

in m

eter

s)

0

0

0

0100

10.0

tsv

10

0

0 1 2 3 4t = time (in seconds)

1 0

1 0

1 0 1

1

1 0 1

0 1.0 1.0* 10.0

90.0* 9.8

19.8

t t ts v t

s s sv g t

v v v

An Illustration (cont.)

100

90

80

70

60

50

40

30

20s = D

ispl

acem

ent (

in m

eter

s)

1

1

1

190.0

19.8

tsv

10

0

0 1 2 3 4t = time (in seconds)

2 1

2 1 1

2 1

1.0 1.0 2.0* 70.2* 29.6

t t ts s v tv v g t

An Illustration (cont.)

100

90

80

70

60

50

40

30

20s = D

ispl

acem

ent (

in m

eter

s)

2

2

2

2.070.2

29.6

tsv

10

0

0 1 2 3 4t = time (in seconds)

3 2

3 2 2

3 2

2.0 1.0 3.0* 40.6* 39.4

t t ts s v tv v g t

An Illustration (cont.)

100

90

80

70

60

50

40

30

20s = D

ispl

acem

ent (

in m

eter

s)

3

3

3 340.6

39.4sv

t

10

0

0 1 2 3 4t = time (in seconds)

4 3

4 3 3

4 3

2.0 1.0 4.0* 1.2* 49.2

t t ts s v tv v g t

Mathmatical Model

Given an initial time, t0, an initial height, H, and an initial velocity, V, generate the time history of heights, sn, and velocities, v, by the formulas

0 0

1 1

1

,,,

1, 2,

n n n n n

n n n n

s H v Vs v t s s sv g t v v v

n

Computational Model

Pseudo CodeInput

t0, initial time; V, initial velocity; H, initial heightg, acceleration due to gravity; Δt, time step;imax, maximum number of steps

Outputti, t-value at time step isi, height at time tivi, velocity at time ti

Example: Falling Rock

Initializeset ti = t0 = 0; vi = v0 = V; si = s0 = Hprint ti, si, vi

Time stepping: i = 1, imaxset ti = ti + Δtset si = si + vi*Δtset vi = vi - g*Δtprint ti, si, viif (si <= 0), quit Excel Model

Interpretation

To create a more more realistic model of a falling rock, some of the simplifying assumptions could be dropped; e.g., incor-porate drag - depends on shape of the rock, is proportional to velocity. Improve discrete model: Approximate velocities in the midpoint of time

intervals instead of the beginning. Reduce the size of Δt.

A Virtual Science Laboratory

The site below is a virtual library to visualize science. It has projects in mechanics, electricity and magnetism, life sciences, waves, astrophysics, and optics. It can be used to motivate the development of mathematical models for computational science projects.

explorelearning

Referenced URLs

AiS Challenge Archive site        www.challenge.nm.org/Archive/

Explorescience sitewww.explorelearning.com

Boeing examplewww.boeing.com/commercial/777family/index.html

Road maps for the human brain and climate modeling exampleswww.itrd.gov/pubs/blue00/hecc.html

An interesting modeling site

Formulating modelswww.cnr.colostate.edu/class_info/nr575/webfiles/

L05_Formulating_Continuous_Time_Models.pdf Falling bodies

http://hypertextbook.com/physics/mechanics/falling/

Master tools http://www.shodor.org/master/